Lattices and Number Theory — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Programme : COMMON PROG GROUP (NEP-UGCF 2022) 1. (a) Define a"totally ordered set. Let P = {1, 2, 3, 4, 6, 9, 12, 18, 36} and x ≤ y in P if x divides y. Prove that P is a partial ordered set under the order relation <. Is (b) Define top and bottom elements of a partial order set. List all the maximal, minimal, all the lower and upper bounds along with top and botton element of the following partial ordered set : (c) "Draw the Hesse Diagrams for the parțial order sets 2 v 4 and 2 × 2 where n 'denote the chain: 2. (a) Define sublattice of a lattice L. Consider the following lattice L : Find three sublattices of the above lattice. Deduce that the union of two sublattices is not a sublattice of L. (b) Write distributive laws in a lattice L. State the criterion for a lattice to be distributive. Is the following lattice distributive? Justify. Find minimal form of the polynomial p. = xy + yby using : Karnaugh maps (a) State the Duality Principle in à partial orderd set. Find the dual of : • (b) Find the DN and CN form of p = x + xy. . ' (C) Define Boolean Algebra. Represent the following circuit diagram in the form of Boolean expression : (a) State Division Algorithm. Use the Euclidean Algorithm to obtain integers x and y satisfying the equation : (b) State Wilson's theorem and prove that : (c) State Chinese Remainder Theorem and obtain eight incongruent solutions of the linear congruence: (a) Define Euler's Phi-Function. Usë Euler's theorem to establish that for any integer 'a' : (b) Define primitive root of a integer n with example. If F = 22" , n> 1 is a prime, then prove that 2 is not a primitive root of Fn (c) Find the order of the integer 3' modulo 19. 6. • (a) Solve the following quadratic congruence : (b) Define quadratic residue of an odd prime p. Find all the quadratic residues of 17. (c) Define Legendre's symbol. Find the value of the following Legendre symbols :